\chapter{Divisors on Toric Varieties}
\label{divisors}

\section{Normality and Divisors}
 
\begin{itemize}
    \item{codimension one is locally principle}
    \item{DVR's, $v_D(f)$}
    \item{Toric divisors, $D = D_\rho$ then $v_D(\chi^u) = \langle u,
       n_\rho \rangle$} 
    \item{example of locally principal but not globally principal}
    \item{support of a divisor}
\end{itemize}

\section{Cartier and Weil Divisors on Toric Varieties}
 
\begin{itemize}
    \item{Define $\cO_Y(U)$ (don't call it a sheaf)}
    \item{Affine torics Cartier iff principal, general condition $\sum
       a_iD_i$ Cartier iff $\forall$ maximal $\sigma$, $\exists m_\sigma
       \in M$ such that $a_i = \langle m_\sigma, n \rangle, \forall n
       \in \sigma$} 
    \item{$T$-invariant divisors}
    \item{$X$ smooth implies Weil = Cartier; Cartier = Weil in
       general implies UFD but for torics implies smooth}
\end{itemize}

\section{Divisor Classes on Toric Varieties}
 
\begin{itemize}
    \item{Class group}
    \item{Picard group}
    \item{Proof $\Cl(X)$ is same as that defined in Chapter 4}
\end{itemize}